Dark-matter dynamical friction versus gravitational-wave emission in the evolution of compact-star binaries L. Gabriel G´omez1,2,3 and J. A. Rueda,1,3,4 1Dipartimento di Fisica and ICRA, Sapienza Universit`adi Roma, P.le Aldo Moro 5, I–00185 Rome, Italy 2University of Nice-Sophia Antipolis, 28 Av. de Valrose, 06103 Nice Cedex 2, France 3ICRANet, Piazza della Repubblica 10, I–65122 Pescara, Italy and 4ICRANet-Rio, CBPF, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290–180, Brazil (Dated: August 15, 2017) The measured orbital period decay of relativistic compact-star binaries, with characteristic or- bital periods ∼ 0.1 days, is explained with very high precision by the gravitational wave (GW) emission of an inspiraling binary in vacuum predicted by general relativity. However, the binary gravitational binding energy is also affected by an usually neglected phenomenon, namely the dark matter dynamical friction (DMDF) produced by the interaction of the binary components with their respective DM gravitational wakes. Therefore, the inclusion of the DMDF might lead to a binary evolution which is different from a purely GW-driven one. The entity of this effect depends on the orbital period and on the local value of the DM density, hence on the position of the binary in the Galaxy. We evaluate the DMDF produced by three different DM profiles: the Navarro-Frenk-White (NFW) profile, the non-singular-isothermal-sphere (NSIS) and the Ruffini-Arg¨uelles-Rueda (RAR) DM profile based on self-gravitating keV fermions. We first show that indeed, due to their Galactic position, the GW emission dominates over the DMDF in the NS-NS, NS-WD and WD-WD binaries for which measurements of the orbital decay exist. Then, we evaluate the conditions (i.e. orbital period and Galactic location) under which the effect of DMDF on the binary evolution becomes comparable to, or overcomes, the one of the GW emission. We find that, for instance for 1.3–0.2 M⊙ NS-WD, 1.3–1.3 M⊙ NS-NS, and 0.25–0.50 M⊙ WD-WD, located at 0.1 kpc, this occurs at orbital periods around 20–30 days in a NFW profile while, in a RAR profile, it occurs at about 100 days. For closer distances to the Galactic center, the DMDF effect increases and the above critical orbital periods become interestingly shorter. Finally, we also analyze the system parameters (for all the DM profiles) for which DMDF leads to an orbital widening instead of orbital decay. All the above imply that a direct/indirect observational verification of this effect in compact-star binaries might put strong constraints on the nature of DM and its Galactic distribution. Keywords: DM: density profiles, velocity distribution function, halo- Galaxy: dynamical friction, Pulsars: Binaries, orbital period decay I. INTRODUCTION of massive black hole (BH) binaries in a stellar medium [6]. Thus, dynamical friction plays an important role in Compact-star binaries composed of neutron stars the orbital evolution of many astrophysical systems. In (NSs) and/or white dwarfs (WDs) have turned out to be a pioneering work, S. Chandrasekhar [7] calculated the rich laboratories of physics and astrophysics that allow dynamical friction force on a massive object traversing to test fundamental theoretical predictions. In particu- an infinite homogeneous collisionless background (repre- lar, NS-NS binaries have served to prove the existence of senting the surrounding star neighbors). gravitational waves (GWs) [1] and the motion of matter It is thus natural to expect that a binary system mov- and photons in the strong gravitational fields [2], as well ing through the galaxy can also experience a dynami- as other phenomena [3]. These latter aspects are of spe- cal friction caused by collisionless DM particles, namely cial interest in tests of general relativity and alternative DM dynamical friction (hereafter DMDF), particularly in arXiv:1706.06801v2 [astro-ph.GA] 13 Aug 2017 theories of gravity [2, 4]. DM-dominated regions, as at the outer part of the Galac- The orbital motion of such systems also offers the tic halo and near the Galactic center [8]. The perturbed possibility of analyzing further effects. An interesting orbital motion may lead thus to interesting observable physical situation arises when the orbiting object moves effects in the secular evolution of the orbital period. An through an extended medium which is formed, for in- interesting proposal was advanced in Ref. [9] on the pos- stance, from the mass loss of the binary companion. This sibility of inferring constraints to the DM density by de- interaction can be thought as a drag force exerted by the termining the above DM effect on the orbital motion of circumbinary medium on the object in question, perturb- binaries (see also the pioneering work by Bekenstein & ing thereby its Keplerian orbital motion [5]. This dynam- Zamir [10], for a general discussion of collisionless back- ical friction produced by the gravitational drag-force has ground types as well as in the context of DM). They been also studied in the context of different astrophysi- showed that the change in the orbital period could be cal phenomena such as mergers of star clusters, galaxies, due to the dynamical friction force exerted by the DM and even galaxy clusters, to the inspiral of dwarf galax- background on the binary. In that work, this effect was ies within dark-matter halos and the orbital evolution used to put an upper bound on the DM density in a given 2 location of the Galaxy, independently of the density pro- pulsar and reproduce some general results presented in file or the nature of the DM particles. It can be shown, [9]. Furthermore, we introduce Galactic-halo observables however, that this upper limit is indeed fulfilled by any in order to generalize the prescription presented in [9] DM density profile consistent with the outer halo prop- and present thus a more realistic estimation of dynami- erties of the Milky Way. Thus, we explore in this work cal friction effects. Finally, we present in section V the the dependence of the orbital period decay by DMDF numerical results of P˙b as a function of the radial posi- on the different binary parameters and also on the DM tion, the DM wind velocity and the orbital period. This density profile, in order to identify all possible physical latter computation leads us to compare directly the P˙b situations suitable for an observational verification of the due to GW emission to that given by DMDF. In sec- DMDF effect. For doing this we obtain DM profiles ful- tion V we summarize our results and present a general filling definite Galactic-halo observables such as the es- discussion. cape velocity, the velocity dispersion and the one-halo scale length parameters. The velocity distribution func- tion and the DM density profile are, as we shall show II. BINARY SYSTEMS AND ORBITAL PERIOD below, crucial elements in the dynamical friction force DECAY BY GRAVITATIONAL WAVES estimation. It is known that the DM in the outer part of our Galaxy The precise pulsar timing measurements allow us to is well described by a classical Maxwell-Boltzmann dis- detect, with a high accuracy, tiny orbital effects which tribution, e.g. by a non-singular isothermal (hereafter thus require a precise theoretical description of the or- NSIS) profile [6]. However, depending on the DM na- bital motion [1]. In the weak field regime (Newtonian ture (e.g. particle type), the DM density distribution approach), the binary motion of pulsar is simply de- can deviate from the classical Maxwell-Boltzmann be- scribed by the Kepler laws. However, relativistic and havior towards the inner regions of the Galaxy. This im- strong-field effects in the orbital motion should be taken plies that the DMDF effect will depend according to the into account in the vicinity of a close-orbit binary pulsar phase-space density consistent with the DM particle na- [2]. These relativistic effects can be described, for the ture. We shall consider, for the sake of comparison, three known binaries, with sufficient accuracy in terms of the DM models: 1) the NSIS profile, 2) the Navarro-Frenk- called post-Keplerian parameters that account for depar- White (NFW) profile [11], and 3) the recently introduced tures from Newtonian Keplerian dynamics owing e.g. to Ruffini-Arg¨uelles-Rueda (RAR) model [12, 13]. the GW emission, time delay caused by the curvature The RAR model is based on a self-gravitating system of space-time near the pulsar (Shapiro delay), and rela- of massive (keV) fermions in thermodynamic equilibrium. tivistic time dilation [16]. There exists a variety of effects The density profile of the RAR model exhibits a core-halo that affect the orbital period stability and they can be, structure which allows to explain the DM distribution in roughly speaking, classified in two large groups: kine- galactic halos from dwarfs to big spirals, and predicts at matic and intrinsic to the system. The former include the same time the presence of a DM high density core the effects of a secular increase due to the Galactic gravi- [12]. Under this approach and following the more re- tational potential, secular acceleration resulting from the alistic distribution function including violent relaxation pulsars transverse velocity (proper motion of the pulsar) processes [14] and the escape velocity of particles, the and the clusters gravitational field; while the latter is re- Fermi-Dirac distribution function was subsequently in- lated to “local” effects in the system as mass loss either troduced to describe the finite size of halos. In the case of from the pulsar or its companion and the GW emission the Milky Way, such a DM core can explain the observed among others2.